Problem 28 - College Algebra with Modeling and Visualization

Question

Use grouping to factor the polynomial. \(8 x^{3}-2 x^{2}+12 x-3\)

Step-by-Step Solution

Answer

The polynomial factors to \((4x - 1)(2x^2 + 3)\).

Step 1: Group the Terms

Start by grouping the polynomial into two pairs. For the polynomial \(8x^3 - 2x^2 + 12x - 3\), group it as \((8x^3 - 2x^2) + (12x - 3)\). This is done to simplify the expression by finding common factors in each group.

Step 2: Factor Out the Greatest Common Factor from Each Group

Now, factor out the greatest common factor from each group. For the first group \((8x^3 - 2x^2)\), the GCF is \(2x^2\), giving \(2x^2(4x - 1)\). For the second group \((12x - 3)\), the GCF is \(3\), giving \(3(4x - 1)\). The expression is now \(2x^2(4x - 1) + 3(4x - 1)\).

Step 3: Factor by Grouping

Notice both groups now contain the common binomial factor \((4x - 1)\). Factor out the \((4x - 1)\), resulting in \((4x - 1)(2x^2 + 3)\).

Step 4: Verify the Factored Form

To ensure that the factorization is correct, expand \((4x - 1)(2x^2 + 3)\) to check if it equals the original polynomial. Doing so, you should get \(8x^3 - 2x^2 + 12x - 3\), confirming that our factorization is accurate.

Key Concepts

Grouping Method

The grouping method is a powerful technique used in factoring polynomials, especially when the polynomial has four or more terms. It involves breaking down a polynomial into smaller groups to find common factors more easily. In the exercise, the polynomial is divided into two pairs:

• The first group is \((8x^3 - 2x^2)\)
• The second group is \((12x - 3)\)

By grouping these terms, it simplifies the process of finding common factors, enabling us to factor the overall polynomial more effectively. The key is to arrange terms such that each pair has a common factor.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more terms without any remainder. In polynomial factorization, finding the GCF is crucial to simplifying expressions.
For the group \((8x^3 - 2x^2)\), identify the GCF as \(2x^2\).

• Divide each term by the GCF to factor it out, resulting in \(2x^2(4x - 1)\).

For the second group \((12x - 3)\), the GCF is \(3\),

• which gives us \(3(4x - 1)\) after factoring.

Recognizing and extracting the GCF simplifies the polynomial, making it easier to identify other shared factors.

Binomial Factor

A binomial factor is a two-term polynomial that can often be factored out of an expression as a common factor. In the given problem, after applying the grouping method and extracting the GCF from each pair, each group contained the common binomial factor \((4x - 1)\).

• This commonality allows us to factor it out of each expression, combining them into one cohesive factorization.

The polynomial was factorized from \(2x^2(4x - 1) + 3(4x - 1)\) to \((4x - 1)(2x^2 + 3)\), streamlining the original polynomial into two terms linked by multiplication, thus easing further mathematical operations or verification.

Verification of Factorization

Verification of factorization is a vital step to ensure that the polynomial has been factored correctly. This can be done by expanding the factored form and checking if it matches the original polynomial.
Starting with the factored expression \((4x - 1)(2x^2 + 3)\), perform the distributive property:

• Multiply \(4x\) by each term in \((2x^2 + 3)\), resulting in \(8x^3 + 12x\).
• Then multiply \((-1)\) by each term, giving \(-2x^2 - 3\).
• Combine these results to verify: \(8x^3 - 2x^2 + 12x - 3\).

Since the expanded form matches the original polynomial, this confirms the factorization is accurate. Verification not only checks accuracy but reinforces the understanding of polynomial structures.